University:
Fresno Pacific University
Course:
MATH 210 | Calculus I
Academic year:

2024
Views:

285

Pages:

2
Author:

Athena Harrison

Name: Key Period: Integration Using Substitution The substitution method turns an unfamiliar integral into one we can evaluate. In other words, substitution gives us a simpler integral involving the variable u. **Five Steps for Integration by Substitution** Step 1: Choose a new variable u Step 2: Determine the value du Step 3: Make the substitution Step 4: Integrate the resulting integral Step 5: Return to the initial variable x **EX: Evaluate** ∫(x²+5)²2x dx Let u = x²+5 du = 2x dx ∫u²du = u³/3 + C → (x²+5)³/3 + C **EX: Evaluate** ∫(1/2x+1) dx u = 2x+1 du = 2 dx ½ du = dx ∫(1/u) * ½ du = ½ ∫(1/u) du = ½ ln|u| + C = ½ ln|2x+1| + C **EX: Evaluate** ∫cos 2x dx u = 2x du = 2 dx ½ du = dx ∫cos u * ½ du = ½ ∫cos u du = ½ sin u + C = ½ sin 2x + C EX: Evaluate ∫x√x²+1 dx u = x²+1 du = 2x dx ½ du = x dx ∫√u * ½ du = ½ ∫u^(1/2) du = ½ * (2/3)u^(3/2) + C = (1/3)u^(3/2) + C = (1/3)(x²+1)^(3/2) + C EX: Evaluate ∫x³e^(x⁴) dx u = x⁴ du = 4x³ dx ¼ du = x³ dx ∫e^u * ¼ du = ¼ ∫e^u du = ¼ e^u + C = ¼ e^(x⁴) + C EX: Evaluate ∫(1/x)ln|x| dx u = ln|x| du = (1/x) dx ∫u du = ln|u| + C = ln|ln|x|| + C EX: Evaluate ∫x√4+x dx Need to replace x dx So use eq to get x = u - 4 u = 4+x du = dx u - 4 = x ∫√u(u-4) du = ∫(u^(3/2) - 4u^(1/2)) du = (2/5)u^(5/2) - (8/3)u^(3/2) + C = (2/5)(4+x)^(5/2) - (8/3)(4+x)^(3/2) + C

15.2 Notes Integration Using Substitution

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